Problem 45 - Calculus: Early Transcendentals

Question

Find the limit, if it exists. If the limit does not exist, explain why. \( \displaystyle \lim_{x \to 0^-}\left(\frac{1}{x} - \frac{1}{|x|} \right) \)

Step-by-Step Solution

Answer

The limit is \(-\infty\).

Step 1: Understand the expression

The expression to find the limit for is \( \frac{1}{x} - \frac{1}{|x|} \). Noting that \(|x|\) is the absolute value of \(x\), when approaching 0 from the left \(x < 0\).

Step 2: Simplify \(|x|\) for \(x < 0\)

For values of \(x\) approaching 0 from the negative side, \(|x| = -x\). This transforms the expression to \( \frac{1}{x} - \frac{1}{-x} \).

Step 3: Simplify the expression

Simplify the expression \( \frac{1}{x} - \frac{1}{-x} \). This becomes \( \frac{1}{x} + \frac{1}{x} = \frac{2}{x} \).

Step 4: Evaluate the limit

Find \( \lim_{x \to 0^-} \frac{2}{x} \). As \(x\) approaches 0 from the left, \(x\) is a small negative number, making \(\frac{2}{x}\) a large negative number. Thus, the limit does not exist and it tends to negative infinity, \(-\infty\).

Step 5: Conclusion

As \(x\) approaches 0 from the left, \(\frac{2}{x}\) becomes more negative without bound. Hence, \(\lim_{x \to 0^-}\left(\frac{1}{x} - \frac{1}{|x|} \right) = -\infty\).

Key Concepts

Limits

When we talk about limits in calculus, we're trying to understand what a function approaches as the input, or variable, gets close to a certain number. Limits help us investigate function behavior near target values, enabling a better grasp of potential infinite behaviors, discontinuities, or asymptotes.

To find a limit, like in our problem \[\lim_{x \to 0^-}\left(\frac{1}{x} - \frac{1}{|x|} \right)\]we explore the behavior of the function as \(x\) approaches 0 from the left, meaning \(x\) is getting closer to zero through negative values. This requires evaluating function expressions carefully and understanding their numerical trends as the input gets closer to the target number.

A key part of solving limits is recognizing whether a function tends to a real number or if it approaches infinity (positive or negative). In our case, the function simplifies in such a way that the limit heads towards negative infinity, \[-\infty\]emphasizing the importance of knowing various mathematical rules and concepts like absolute value, negative infinity, and simplification when working with limits.

Negative infinity

Negative infinity, represented as \[-\infty\]indicates a boundless decrease without end. In mathematical terms, it means you are heading towards an infinitely large negative number. This concept is crucial for understanding and solving limit problems involving expressions that decrease indefinitely, like fractions where the denominator approaches zero from the left.

In our exercise, when we deduce the simplified expression to \[\frac{2}{x}\]we analyze it as \(x\) tends towards zero from the negative side. Because we're dividing by a small negative number, the result becomes a large negative value which keeps growing as \(x\) approaches zero. Therefore, the limit tends towards negative infinity, meaning the values of the function decrease indefinitely from the left side approach to zero.

Understanding negative infinity is essential, especially when dealing with limit expressions involving division by variable approaching zero, ensuring clarity on how values trend towards negative values infinitely.

Absolute value

The absolute value of a number, denoted by \(|x|\)represents the distance of that number from zero on the number line, ignoring the direction. Essentially, it turns positive any number by stripping away negative signs.

In limits and other calculus calculations, understanding absolute value is crucial as functions behave differently depending on whether inputs are positive or negative. In our original exercise,\[\lim_{x \to 0^-}\left(\frac{1}{x} - \frac{1}{|x|} \right)\]it's key to comprehend how absolute value affects \(x\). Here, since \(x\) is less than zero, \(|x| = -x\)which is pivotal in transforming the limit expression properly into terms we can more directly evaluate.

Recognizing how absolute value affects expressions, especially in limits, facilitates proper simplification, ensuring we capture all changes in signs and values correctly at approaching points. This concept helps tease apart behavior on either side of the number line, especially when approaching zero.